Is Hijab an obligation? Where is it mentioned in the Qur-aan? What is the Hijab?

Is Hijab an obligation? Where is it mentioned in the Qur-aan?
What is the Hijab?

Unfortunately the question isn't really answered comprehensively and many Muslims, especially sisters, are left very confused.
This topic is sensitive, but Allah azzawajal is our Lord and Protector and whatever we do, we do to please Him.

Hijab is mentioned in the Qur'an. However, it is mentioned not in terms of a dress or modesty, but rather as a 'veil' or 'partition.'
So for example, Allah says in Surah Sa'ad that He hid the sun behind a 'hijab.'
So hijab is not really mentioned in the Qur'an in terms of dressing.

So then what does the Qur'an say about the physical dress?
The answer is broken into four parts:

- Khimar: In surah An-Nur, verse 31:
Allah says "And to draw their bikhomoorihinna over their chests area."

Khimar in the classic Arabic language is a headscarf that covers the area around the head and flows down with the hair. Allah commanded the women to wear the khimar and draw it around the chest area, so as to form the complete headscarf that we see today for sisters.

- Jilbaab: In surah Al-Ahzab, verse 59 Allah says "O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves their jalabeebihinna."

Jilbaab in the classic Arabic language refers to a one-piece garment that extends from the shoulders to the feet, and goes over the normal clothes you wear when you are in public (like the way you wear a winter coat when going into the cold on top of your normal clothes). Allah commanded women to wear this when going out.

- Tabarruj: In surah Al-Ahzab, verse 33 Allah says "And abide in your houses and do not tabarrajna yourselves as [was] the tabarruja of the former times of ignorance."

Tabarruj is a word meaning those things that attract attention. It is putting out for display to draw the attention of others. As an example, if all the people in a society wear black clothes and a person wears white clothes, that person is doing tabarruj as that person is drawing the gazes of people to them.

- Zeenah: In surah An-Nur, in verse 31 Allah says "And tell the believing women to reduce [some] of their vision and guard their private parts and not expose their zeenatahunna except that which is apparent."

Zeenah are adornments that one wears. The verse gives an example at the end of it by saying "And let them not stamp their feet to make known what they conceal of their zeenatihinna." The example attached to this mentions the anklets women would wear and stamp their feet so they would be noticed.

These are the four parts Allah Himself mentions in the Qur'an.
As we observe human beings we realize that when left to their own devices, men naturally exploit the physical reality of women. It is taken as product in the market, with each product having differing values. This is why models are 'devalued' in the industry as time progresses, because they no longer pretty enough due to aging. Allah recognizes human beings based on their intellectual merit, and their piety and not on their physicality. We are all human beings who have our own thoughts, our own personality, and those are the key parts of who we are.